Question

find a and b rational number
plzzz plzzz anyone else
i will mark as brainliest ⭐⭐⭐⭐⭐⭐
30 points....

Answer 1

Hey friend,
Here is the answer you were looking for:
$\frac{2 - \sqrt{5} }{2 + \sqrt{5} } = a \sqrt{5} + b$

We can write it as,

$= \frac{2 - \sqrt{5} }{2 + \sqrt{5} } = a \sqrt{5} + b$

On rationalizing the denominator we get,

$= \frac{2 - \sqrt{5} }{2 + \sqrt{5} } \times \frac{2 - \sqrt{5} }{2 - \sqrt{5} }$

Using the identities :

${(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy \\ (x + y)(x - y) = {x}^{2} - {y}^{2}$

$= \frac{ {(2)}^{2} + {( \sqrt{5} )}^{2} - 2(2)( \sqrt{5} )}{ {(2)}^{2} - {( \sqrt{5} )}^{2} } \\ \\ = \frac{4 + 5 - 4 \sqrt{5} }{4 - 5} \\ \\ = \frac{9 - 4 \sqrt{5} }{ - 1} \\ \\ = - (9 - 4 \sqrt{5} ) \\ \\ = - 9 + 4 \sqrt{5} \\ \\ 4 \sqrt{5} - 9 = a \sqrt{5} + b \\ \\ a = 4 \: : \: b = - 9$

Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
☺☺

Answer 2

⭐⭐Hello friend..Ur answer is here⤵⤵

$\frac{2 - \sqrt{5} }{2 + \sqrt{5} } \\ \\ multiplying \: by \: 2 - \sqrt{5} in \: numerator \: and \: dinominator \\ \\ \frac{ {(2 - \sqrt{5} )}^{2} }{4 - 5} \\ \\ \frac{4 + 5 - 4 \sqrt{5} }{ - 1} \\ \\ \frac{9 - 4 \sqrt{5} }{ - 1} \\ \\ - 9 + 4 \sqrt{5} \\ \\ 4 \sqrt{5 } - 9 \\ \\ compare \: with \: a \sqrt{5} + b \: \: we \: get \\ \\ a = 4 \: \: and \: b = - 9$

I HOPE IT IS HELPFUL TO YOU ☺